If you are deploying a Silverlight 4-application it has to run out of the browser as the sandbox inside the browser does not allow ElevatedPermissions. If you decide to use Silverlight 5 you can enable ElevatedPermissions with an administrative account for in-browser apps (msdn.microsoft.com/en-us/library/ee721083(v=vs.95).aspx).

Though you can of course run the app inside the intranet-page it has no elevated permissions as long as you (or the user) don't install and run it OOB. Thus the app will not be able to access the username.
It seems this restriction is partially gone with Silverlight 5 but I have not tested this, yet and cannot tell you if you could go the way described above.

I can only guess how you're going to use the username, but if you're planning to use it for authentication or authorization, you should do a bit more research.

If you want to actually authenticate the user, the server must be the one to do so. eg: asking for a username & password, windows authentication, etc. The username (and even the domain) that's running on a PC shouldn't be trusted to be correct.